Shore’s computational reverse mathematics Benedict Eastaugh University of Bristol Philosophy and Computation workshop Lunds universitet May 13, 2012 Benedict Eastaugh (University of Bristol) Shore’s computational reverse mathematics May 13, 2012 1 / 26
Reverse mathematics and foundational commitments 1 Computational reverse mathematics 2 Computable entailment and justification 3 Benedict Eastaugh (University of Bristol) Shore’s computational reverse mathematics May 13, 2012 2 / 26
A foundational dialectic Suppose we’re committed to a particular foundational programme of limited strength, such as predicativism or finitistic reductionism. Benedict Eastaugh (University of Bristol) Shore’s computational reverse mathematics May 13, 2012 3 / 26
A foundational dialectic Suppose we’re committed to a particular foundational programme of limited strength, such as predicativism or finitistic reductionism. 1 How do we know which theorems we’re entitled to assert? Benedict Eastaugh (University of Bristol) Shore’s computational reverse mathematics May 13, 2012 3 / 26
A foundational dialectic Suppose we’re committed to a particular foundational programme of limited strength, such as predicativism or finitistic reductionism. 1 How do we know which theorems we’re entitled to assert? 2 How do we know what mathematics we’re giving up? Benedict Eastaugh (University of Bristol) Shore’s computational reverse mathematics May 13, 2012 3 / 26
Reverse mathematics can help If we formalise our foundation in second order arithmetic , results in reverse mathematics will let us know which theorems we’re entitled to assert and which remain out of reach. Benedict Eastaugh (University of Bristol) Shore’s computational reverse mathematics May 13, 2012 4 / 26
Reverse mathematics can help If we formalise our foundation in second order arithmetic , results in reverse mathematics will let us know which theorems we’re entitled to assert and which remain out of reach. This is done by proving equivalences between such theorems and subsystems of second order arithmetic, over a weak base theory. Benedict Eastaugh (University of Bristol) Shore’s computational reverse mathematics May 13, 2012 4 / 26
Syntax and semantics of second order arithmetic Second order arithmetic is a two-sorted first order system with number variables m , n , i , j , . . . and set variables X , Y , Z , . . . ranging over subsets of the domain. Benedict Eastaugh (University of Bristol) Shore’s computational reverse mathematics May 13, 2012 5 / 26
Syntax and semantics of second order arithmetic Second order arithmetic is a two-sorted first order system with number variables m , n , i , j , . . . and set variables X , Y , Z , . . . ranging over subsets of the domain. L 2 -structures are models of the first order language of arithmetic extended with a collection of sets for the second order variables to range over: M = � M , S , + , · , <, 0 , 1 � where S ⊆ P ( M ). Benedict Eastaugh (University of Bristol) Shore’s computational reverse mathematics May 13, 2012 5 / 26
Axioms of second order arithmetic The axioms of second order arithmetic or Z 2 are the universal closures of the following: Basic arithmetic axioms: PA minus induction. Benedict Eastaugh (University of Bristol) Shore’s computational reverse mathematics May 13, 2012 6 / 26
Axioms of second order arithmetic The axioms of second order arithmetic or Z 2 are the universal closures of the following: Basic arithmetic axioms: PA minus induction. Induction axiom: (0 ∈ X ∧ ∀ n ( n ∈ X → n + 1 ∈ X )) → ∀ n ( n ∈ X ) . Benedict Eastaugh (University of Bristol) Shore’s computational reverse mathematics May 13, 2012 6 / 26
Axioms of second order arithmetic The axioms of second order arithmetic or Z 2 are the universal closures of the following: Basic arithmetic axioms: PA minus induction. Induction axiom: (0 ∈ X ∧ ∀ n ( n ∈ X → n + 1 ∈ X )) → ∀ n ( n ∈ X ) . Comprehension scheme: ∃ X ∀ n ( n ∈ X ↔ ϕ ( n )) for all ϕ with X not free. Benedict Eastaugh (University of Bristol) Shore’s computational reverse mathematics May 13, 2012 6 / 26
Subsystems of second order arithmetic Subsystems of Z 2 are primarily obtained by restricting the comprehension scheme to particular syntactically defined subclasses. Subsystems of Z 2 Defining conditions Recursive (∆ 0 RCA 0 1 ) comprehension WKL 0 RCA 0 plus weak K¨ onig’s lemma ACA 0 Arithmetical comprehension ATR 0 ACA 0 plus arithmetical transfinite recursion Π 1 Π 1 1 − CA 0 1 comprehension Benedict Eastaugh (University of Bristol) Shore’s computational reverse mathematics May 13, 2012 7 / 26
Foundational programmes and the Big Five The most important subsystems of second order arithmetic, known as the Big Five, formally capture some philosophically-motivated programmes in foundations of mathematics. Benedict Eastaugh (University of Bristol) Shore’s computational reverse mathematics May 13, 2012 8 / 26
Foundational programmes and the Big Five The most important subsystems of second order arithmetic, known as the Big Five, formally capture some philosophically-motivated programmes in foundations of mathematics. Subsystems of Z 2 Foundational programmes Constructivism RCA 0 Finitistic reductionism WKL 0 Predicativism ACA 0 Predicative reductionism ATR 0 Π 1 Impredicativity 1 − CA 0 Benedict Eastaugh (University of Bristol) Shore’s computational reverse mathematics May 13, 2012 8 / 26
Varieties of induction The second order induction axiom ties the strength of induction to the strength of the comprehension axiom: we can do induction only over those sets we can prove exist. Benedict Eastaugh (University of Bristol) Shore’s computational reverse mathematics May 13, 2012 9 / 26
Varieties of induction The second order induction axiom ties the strength of induction to the strength of the comprehension axiom: we can do induction only over those sets we can prove exist. Contrast this with the induction scheme , each instance of which is a theorem of Z 2 : ( ϕ (0) ∧ ∀ n ( ϕ ( n ) → ϕ ( n + 1))) → ∀ n ϕ ( n ) for all formulae ϕ in the language of second order arithmetic. Benedict Eastaugh (University of Bristol) Shore’s computational reverse mathematics May 13, 2012 9 / 26
Varieties of induction The second order induction axiom ties the strength of induction to the strength of the comprehension axiom: we can do induction only over those sets we can prove exist. Contrast this with the induction scheme , each instance of which is a theorem of Z 2 : ( ϕ (0) ∧ ∀ n ( ϕ ( n ) → ϕ ( n + 1))) → ∀ n ϕ ( n ) for all formulae ϕ in the language of second order arithmetic. Weaker forms of induction can be obtained by restricting this scheme to particular classes such as the Σ 0 1 formulae. Benedict Eastaugh (University of Bristol) Shore’s computational reverse mathematics May 13, 2012 9 / 26
Induction axioms and subsystems of Z 2 Σ 0 1 induction Induction axiom Full induction scheme RCA 0 RCA WKL 0 WKL ACA 0 ACA ATR 0 ATR Π 1 Π 1 1 − CA 0 1 − CA Benedict Eastaugh (University of Bristol) Shore’s computational reverse mathematics May 13, 2012 10 / 26
Reverse mathematics and foundational commitments 1 Computational reverse mathematics 2 Computable entailment and justification 3 Benedict Eastaugh (University of Bristol) Shore’s computational reverse mathematics May 13, 2012 11 / 26
Computational reverse mathematics Developed by Richard Shore in two recent papers (Shore 2010, 2011), computational reverse mathematics draws on recursion theory rather than proof theory. Benedict Eastaugh (University of Bristol) Shore’s computational reverse mathematics May 13, 2012 12 / 26
Computational reverse mathematics Developed by Richard Shore in two recent papers (Shore 2010, 2011), computational reverse mathematics draws on recursion theory rather than proof theory. It has a two-fold motivation: Giving an account of reverse mathematics which most mathematicians will find natural, in computational and construction-oriented terms. Benedict Eastaugh (University of Bristol) Shore’s computational reverse mathematics May 13, 2012 12 / 26
Computational reverse mathematics Developed by Richard Shore in two recent papers (Shore 2010, 2011), computational reverse mathematics draws on recursion theory rather than proof theory. It has a two-fold motivation: Giving an account of reverse mathematics which most mathematicians will find natural, in computational and construction-oriented terms. Extending reverse mathematical analysis from countable structures to uncountable ones. Benedict Eastaugh (University of Bristol) Shore’s computational reverse mathematics May 13, 2012 12 / 26
The main question Can computational reverse mathematics be used to carry out the foundational analysis outlined at the beginning? Benedict Eastaugh (University of Bristol) Shore’s computational reverse mathematics May 13, 2012 13 / 26
The main question Can computational reverse mathematics be used to carry out the foundational analysis outlined at the beginning? To answer this, we first need to look at the details of Shore’s programme. Benedict Eastaugh (University of Bristol) Shore’s computational reverse mathematics May 13, 2012 13 / 26
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